Proposition 75.10.3. Let $k$ be a field. Let $G$ be a group algebraic space over $k$. If $G$ is separated, then $G$ is a scheme.

Proof. This lemma generalizes Lemma 75.10.2 (which covers all cases one cares about in practice). The proof is very similar to the proof of Spaces over Fields, Lemma 68.10.7 used in the proof of Lemma 75.10.2 and we encourage the reader to read that proof first.

By Lemma 75.10.1 the base change $G_{\overline{k}}$ is a scheme. Let $K/k$ be a purely transcendental extension of very large transcendence degree. By Spaces over Fields, Lemma 68.10.5 it suffices to show that $G_ K$ is a scheme. Let $K^{perf}$ be the perfect closure of $K$. By Spaces over Fields, Lemma 68.10.1 it suffices to show that $G_{K^{perf}}$ is a scheme. Let $K \subset K^{perf} \subset \overline{K}$ be the algebraic closure of $K$. We may choose an embedding $\overline{k} \to \overline{K}$ over $k$, so that $G_{\overline{K}}$ is the base change of the scheme $G_{\overline{k}}$ by $\overline{k} \to \overline{K}$. By Varieties, Lemma 32.14.2 we see that $G_{\overline{K}}$ is a Jacobson scheme all of whose closed points have residue field $\overline{K}$.

Since $G_{\overline{K}} \to G_{K^{perf}}$ is surjective, it suffices to show that the image $g \in |G_{K^{perf}}|$ of an arbitrary closed point of $G_{\overline{K}}$ is in the schematic locus of $G_ K$. In particular, we may represent $g$ by a morphism $g : \mathop{\mathrm{Spec}}(L) \to G_{K^{perf}}$ where $L/K^{perf}$ is separable algebraic (for example we can take $L = \overline{K}$). Thus the scheme

\begin{align*} T & = \mathop{\mathrm{Spec}}(L) \times _{G_{K^{perf}}} G_{\overline{K}} \\ & = \mathop{\mathrm{Spec}}(L) \times _{\mathop{\mathrm{Spec}}(K^{perf})} \mathop{\mathrm{Spec}}(\overline{K}) \\ & = \mathop{\mathrm{Spec}}(L \otimes _{K^{perf}} \overline{K}) \end{align*}

is the spectrum of a $\overline{K}$-algebra which is a filtered colimit of algebras which are finite products of copies of $\overline{K}$. Thus by Groupoids, Lemma 38.7.13 we can find an affine open $W \subset G_{\overline{K}}$ containing the image of $g_{\overline{K}} : T \to G_{\overline{K}}$.

Choose a quasi-compact open $V \subset G_{K^{perf}}$ containing the image of $W$. By Spaces over Fields, Lemma 68.10.2 we see that $V_{K'}$ is a scheme for some finite extension $K'/K^{perf}$. After enlarging $K'$ we may assume that there exists an affine open $U' \subset V_{K'} \subset G_{K'}$ whose base change to $\overline{K}$ recovers $W$ (use that $V_{\overline{K}}$ is the limit of the schemes $V_{K''}$ for $K' \subset K'' \subset \overline{K}$ finite and use Limits, Lemmas 31.4.11 and 31.4.13). We may assume that $K'/K^{perf}$ is a Galois extension (take the normal closure Fields, Lemma 9.16.3 and use that $K^{perf}$ is perfect). Set $H = \text{Gal}(K'/K^{perf})$. By construction the $H$-invariant closed subscheme $\mathop{\mathrm{Spec}}(L) \times _{G_{K^{perf}}} G_{K'}$ is contained in $U'$. By Spaces over Fields, Lemmas 68.10.3 and 68.10.4 we conclude. $\square$

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